{"paper":{"title":"On representations of positive integers by $(a+c)^{1/3}x + (b+d)y$, $(a+c)x + \\bigl(k(b+d) \\bigr)^{1/3} y$, and $\\bigl(k(a+c) \\bigr)^{1/3} x + l(b+d) y$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Mohamed El Bachraoui","submitted_at":"2012-10-13T13:58:06Z","abstract_excerpt":"We use sums of Liouville type to count the number of ways a positive integer can be represented by the forms\n  $(a+c)^{1/3}x + (b+d)y$, $(a+c)x + \\bigl(k(b+d) \\bigr)^{1/3} y$, and $\\bigl(k(a+c) \\bigr)^{1/3} x + l(b+d) y$ for nonnegative integers $a,b,c,d,k,l,x,y$ under certain relative primality conditions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.3708","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}